Abstract

Tachyonic spectral densities of ultra-relativistic electron
populations are fitted to the γ-ray spectra of two
TeV blazars, the BL Lacertae objects 1ES 0229+200 and 1ES 0347-121. The
spectral maps are compared to Galactic TeV sources, the γ-ray
binary LS 5039 and the supernova remnant W28. In contrast to TeV
photons, the extragalactic tachyon flux is not attenuated by
interaction with the cosmic background light; there is no absorption of
tachyonic γ-rays via pair creation, as tachyons do
not interact with infrared background photons. The curvature of the
observed γ-ray spectra is intrinsic, caused by the
Boltzmann factor of the electron densities, and reproduced by a
tachyonic cascade fit. In particular, the curvature in the spectral map
of the Galactic microquasar is more pronounced than of the two
extragalactic γ-ray sources. Estimates of the
thermodynamic parameters of the thermal or, in the case of supernova
remnant W28, shock-heated nonthermal electron plasma generating the
tachyon flux are obtained from the spectral fits.

1. Introduction

The goal is to point out evidence for superluminal γ-rays
from two active galactic nuclei, the BL Lacertae objects 1ES 0229+200,
cf. Refs. [1], [2], [3] and [4], and 1ES
0347-121, cf. Refs. [5], [6] and [7]. Spectral
maps of these TeV blazars have recently been obtained by means of
ground-based imaging air Cherenkov detectors [4] and [7]. Here, a
tachyonic cascade fit is performed to the γ-ray
spectrum of these blazars. In contrast to electromagnetic γ-rays,
tachyons cannot interact with infrared background photons, so that
there is no attenuation of the extragalactic tachyon flux by
electron–positron pair production. The observed spectrum is already the
intrinsic one without any need of absorption correction as required in
electromagnetic spectral fits [8] and [9]. The
spectral curvature is generated by the Boltzmann factor of the thermal
electron plasma in the galactic nuclei.

Tachyonic cascade spectra are obtained by averaging the
superluminal spectral densities of individual electrons with
ultra-relativistic Fermi distributions. We use the averaged radiation
densities to perform spectral fits to the γ-ray
spectra of the mentioned BL Lacertae objects as well as the microquasar
LS 5039 and the supernova remnant W28. Tachyonic cascade spectra
generated by thermal electron populations in the active galactic nuclei
(AGNs) provide excellent fits to the observed γ-ray
flux. The temperature, source count, and internal energy of the
ultra-relativistic electron plasma are obtained from the spectral fits.
The spectral map of the Galactic microquasar is no less curved than the
cascades in the AGN spectra. Moreover, the spectral curvature of the
AGNs does not increase with increasing redshift, which is further
evidence for an unattenuated extragalactic γ-ray
flux.

The tachyonic radiation field is a Proca field with negative
mass-square,

(1)

coupled to an electron current ,
where mt
is the mass of the superluminal Proca field Aα,
and q the tachyonic charge carried by the
subluminal current [10], [11] and [12]. The mass
term in (1) is added
with a positive sign, and the sign convention for the metric is diag(−1,1,1,1),
so that
is the negative mass-square of the radiation field. The sign convention
for the field tensor is Fμν=Aν,μ−Aμ,ν.
The negative mass-square refers to the radiation field rather than the
current, in contrast to the traditional approach based on superluminal
source particles emitting electromagnetic radiation [13] and [14]. The
superluminal radiation field does not carry any kind of charge,
tachyonic charge q is a property of subluminal
particles, as is electric charge [15]. The field
equations derived from (1) read ,
and can equivalently be written as Proca equation, ,
subject to the Lorentz condition ,
which follows from current conservation .
An estimate of the tachyon–electron mass ratio obtained from hydrogenic
Lamb shifts is mt/m≈1/238[16].

In Section 2, we further
elaborate on the tachyonic Proca equation by comparing to
electromagnetic theory, and assemble the tachyonic spectral averages
employed in the fits, which are based on the transversal and
longitudinal radiation densities generated by a free electronic spinor
current. In Section 3, the spectral
fitting is explained, and intrinsic spectral curvature is argued by
comparing the cascade spectra of the above-mentioned Galactic and
extragalactic TeV sources. The conclusions are summarized in Section 4.

The Lagrangian (1) resembles
its electrodynamic counterpart, but the negative mass-square of the
Proca field causes striking differences. Apart from the superluminal
speed of the tachyonic quanta, the radiation is partially
longitudinally polarized [12], the gauge
freedom is broken, and freely propagating charges can radiate
superluminal quanta. The analogy to Maxwell's theory becomes even more
transparent in 3D. The tachyonic E and B
fields are related to the vector potential Aα=(A0,A)
by E=A0−∂A/∂t
and B=rotA,
and the field equations decompose into

(2)

where we identified jμ=(ρ,j).
The vector potential is completely determined by the current and the
field strengths; there is no gauge freedom owing to the tachyon mass.
Another major difference to electromagnetic theory is the emergence of
longitudinal wave propagation. To see this, we consider monochromatic
waves, ,
and analogously for the scalar potential and the field strengths, with
amplitudes ,
and ,
respectively. We write k=k(ω)k0,
with a unit wave vector k0.
On substituting this ansatz into the free field equations (2), we find the
dispersion relation .
The transversality condition on the vector potential is ,
so that the remaining transversal amplitudes are ,
and .
If the product
does not vanish, the modes must be longitudinal, satisfying .
In this case, ,
and ,
so that a longitudinal plane wave has no magnetic component.

The spectral fits in Section 3 are based on
the quantized tachyonic radiation densities

(3)

generated by the spinor current
of a uniformly moving electron [11]. The
superscripts T and L refer to the transversal/longitudinal polarization
components defined by
and ΔL=0.
γ is the electronic Lorentz factor, αq
the tachyonic fine structure constant, and mt
the tachyon mass. In the γ-ray broadband fit of
supernova remnant W28, the radiation frequencies ω
stretch over five decades, cf. Section 3. The units =c=1
can easily be restored. We use the Heaviside–Lorentz system, so that αq=q2/(4πc)≈1.0×10−13
and mt≈2.15
keV/c2; the
tachyon–electron mass ratio is mt/m≈1/238.
These estimates are obtained from Lamb shifts of hydrogenic ions [16]. A spectral
cutoff occurs at

(4)

Only frequencies in the range 0ωωmax(γ)
can be radiated by a uniformly moving electron, the tachyonic spectral
densities pT,L(ω,γ)
being cut off at the break frequency ωmax.
A positive ωmax(γ)
requires Lorentz factors exceeding the threshold μt
in (4), since ωmax(μt)=0.
The lower threshold on the speed of the electron for radiation to occur
is thus υmin=mt/(2mμt).
The tachyon–electron mass ratio gives υmin/c≈2.1×10−3,
which is roughly the speed of the Galaxy in the microwave background [17].

The radiation densities (3) refer to a
single spinning charge with Lorentz factor γ; we
average them with a Fermi power-law distribution [18] and [19],

(5)

The particle number is found as ,
where γ1
is the lower edge of Lorentz factors in the source population. m
is the electron mass, and the exponential cutoff is related to the
electron temperature by β=m/(kT).
The power-law exponent δ ranges in a narrow
interval in astrophysical spectral averages [20], [21], [22], [23], [24], [25], [26], [27] and [28], which are
usually done in the classical limit (13) with 0δ4.
Otherwise, there are no conceptual constraints on the power-law index
of these stationary non-equilibrium distributions. Thermal equilibrium
is recovered with δ=0
and γ1=1;
the spectral fits of the blazars and the microquasar in Fig. 1, Fig. 2 and Fig. 3 are
performed with equilibrium distributions. The shocked electron plasma
of supernova remnant W28 requires δ≈3.6
to fit the steep power-law slope in Fig. 4. The
exponent
in (5) defines the
fugacity ;
we use a hat to distinguish
from the electron index customarily defined as α=δ−2,
cf. (13). The
fugacity is related to the chemical potential μ by .

Fig. 1. Spectral map of the BL Lac 1ES 0229+200. HESS
data points from [4]. The solid
line T+L depicts the unpolarized differential tachyon flux dNT+L/dE,
obtained by adding the flux densities ρ1,2
of two electron populations, cf. (16) and Table 1. The
transversal (T, dot-dashed) and longitudinal (L, double-dot-dashed)
flux densities add up to the total unpolarized flux T+L. The
exponential decay of the cascades ρ1,2
sets in at about Ecut≈(mt/m)kT,
implying cutoffs at 3.6 TeV for the ρ1
cascade and 120 GeV for ρ2.

Fig. 2. Spectral map of the BL Lac 1ES 0347-121. HESS
points from [7]. The
unpolarized spectral fit T+L is based on the electron distributions
recorded in Table 1, the
polarized flux components are labeled T and L. The ρ1
cascade is cut at Ecut≈4.0
TeV, and ρ2
at 190 GeV. The curvature in the spectral slope of 1ES 0347-121 at z≈0.188
is less pronounced than of 1ES 0229+200 at z≈0.140,
suggesting that the shape of the rescaled flux density
is intrinsic rather than generated by intergalactic absorption.

Fig. 3. Spectral map of the γ-ray
binary LS 5039 close to periastron. HESS data points at the superior
conjunction [32]. Notation
as in Fig. 1. The
cascade ρ1
is exponentially cut at Ecut≈6.3
TeV, and ρ2
at 190 GeV, cf. Table 1. The
spectral map of this microquasar at a distance of 2.5 kpc is more
strongly curved than of the AGNs in Fig. 1 and Fig. 2,
indicating that the curvature of the AGN spectra is intrinsic as well,
generated by the Boltzmann factor of the thermal electron populations,
cf. caption to Fig. 2.

Fig. 4. γ-Ray broadband of the TeV
source HESS J1801-233 and the associated EGRET source 3EG J1800-2338 at
the northeast boundary of supernova remnant W28. Data points from [35], also see [36] and [37]. Notation
as in Fig. 1. The
nonthermal cascade ρ1
admits a power-law slope ∝E1−α,
α≈1.6, adjacent to the MeV–GeV plateau typical for
tachyonic cascade spectra [11] and [12]. A spectral
break at mtγ1≈5.8
GeV is visible as edge in the longitudinal component. The
curvature of the thermal cascade ρ2
in the MeV range is due to the exponential cutoff at (mt/m)kT≈20
MeV, cf. Table 1.

where θ is the Heaviside step function. The
spectral range of densities (3) is bounded
by ωmax
in (4), so that the
solution of
defines the minimal electronic Lorentz factor for radiation at this
frequency [29],

which separates the spectrum into a low- and high-frequency band [30]. By making
use of the spectral functions (8), we can
write the averaged radiation densities (6) as

(10)

with
in (7) and ω1
in (9), so that .
The superscripts T and L denote the transversal and longitudinal
radiation components, cf. (3). The
spectral functions FT,L(ω,γ1)
in (10) are
obtained by substituting radiation densities (3) into the
integral representation (8),

The quasiclassical fugacity expansion of the spectral
functions (11) is found by
expanding density (5) in ascending
powers of .
In leading order, dρF(γ)dρα,β(γ),

(13)

This power-law density is the classical limit of the fermionic density dρF.
As mentioned above, exponent
in the normalization factor Aα,β
defines the fugacity, and is not to be confused with the electron index
α=δ−2
in (13). The
leading order of the reduced spectral functions (12) can be
written as incomplete Γ-function, fk(γ1)Aα,ββδ−kΓ(k−δ,βγ1),
obtained by replacing the fermionic dρF
in the weights (12) by the
Boltzmann power-law density (13).

The classical limit of the fermionic spectral functions FT,L(ω,γ1)
in (8) is the
Boltzmann average [31]

(14)

Here,
is the classical tachyonic spectral density, recovered by dropping all
terms containing mt/m
ratios in (3) and (4). In
particular,
and ,
cf. (3). The
classical spectral functions BT,L(ω,γ1)
are obtained from the Fermi functions FT,L(ω,γ1)
in (11) by dropping
the mt/m
terms, and substituting the leading order of the fugacity expansion of
the weights fk(γ1)
as stated after (13). The
classical limit of the fermionic spectral average pT,L(ω)F
in (10) thus reads

The spectral fits of the active galactic nuclei (AGNs) and the
Galactic TeV sources in Fig. 1, Fig. 2, Fig. 3 and Fig. 4 are based
on the E2-rescaled
flux densities

(16)

where d is the distance to the source and pT,L(ω)α,β
the spectral average (15) (with ω=E/).
The fits are done with the unpolarized flux density dNT+L=dNT+dNL
of two electron populations ρi=1,2,
cf. Table 1. Each
electron density generates a cascade ρi,
and the wideband comprises two cascade spectra labeled ρ1
and ρ2
in the figures. As for the electron count, ,
we use a rescaled parameter
for the fit,

(17)

which is independent of the distance estimate in (16). Here,
implies the tachyon mass in keV units, that is, we put mt≈2.15
in the spectral densities (3). At γ-ray
energies, only a tiny αq/αe-fraction
(the ratio of tachyonic and electric fine structure constants) of the
tachyon flux is absorbed by the detector, which requires a rescaling of
the electron count n1,
so that the actual number of radiating electrons is ,
cf. Ref. [12]. We thus
find the electron count as ,
where
defines the tachyonic flux amplitude extracted from the spectral fit.
(In Table 1, the
subscript 1 of
and
has been dropped.) Electron temperature and cutoff parameter in the
Boltzmann factor of density (13) are related
by kT[TeV]≈5.11×10−7/β,
and the energy estimates of the thermal cascades in Table 1 are
based on ,
cf. Ref. [18]. (The
renormalized count
is to be identified with the particle number N in
the thermodynamic functions discussed in this reference.) The distance
estimates of the AGNs are based on dcz/H0,
with the Hubble distance c/H0≈4.4×103
Mpc (that is, h0≈0.68).
Hence, d[Mpc]≈4.4×103z,
and ,
cf. Table 1.

Table 1.

Electronic source distributions ρi
generating the tachyonic cascade spectra of the active galactic nuclei
in Fig. 1 and Fig. 2, the
microquasar LS 5039 in Fig. 3, and the
supernova remnant W28 in Fig. 4. Each ρi
stands for a thermal Maxwell–Boltzmann density dρα=−2,β(γ)
with γ1=1,
apart from the ρ1
distribution of SNR W28, which is a power-law density with α≈1.6
and γ1≈2.7×106,
cf. (13) and after (5). β
is the cutoff parameter in the Boltzmann factor.
determines the amplitude of the tachyon flux generated by the electron
density ρi,
from which the electron count ne
is inferred at the indicated distance d, cf. after (17). kT
is the temperature and U the internal energy of the
electron populations, cf. after (16). The
parameters β and
are extracted from the least-squares fit T+L in Fig. 1, Fig. 2, Fig. 3 and Fig. 4

Fig. 1 shows the
tachyonic spectral map of the BL Lacertae object (BL Lac) 1ES 0229+200,
located at a redshift of z≈0.140,
cf. Refs. [1], [2], [3] and [4]. The flux
points were obtained with the HESS array of atmospheric Cherenkov
telescopes in the Khomas Highland of Namibia [4]. The χ2-fit
is done with the unpolarized tachyon flux T+L, and
subsequently split into transversal and longitudinal components. The
differential flux is rescaled with E2
for better visibility of the spectral curvature. Temperature and source
count of the electron populations generating the cascades are recorded
in Table 1. In Fig. 2, we show
the spectral map of the BL Lac 1ES 0347-121, at a redshift of z≈0.188,
cf. Refs. [5], [6] and [7]. TeV γ-ray
spectra of BL Lacs are usually assumed to be generated by inverse
Compton scattering or proton–proton scattering followed by pion decay [8]. Both
mechanisms result in a flux of TeV photons, assumed to be partially
absorbed by interaction with infrared background photons owing to pair
creation, so that the intrinsic spectrum has to be reconstructed on the
basis of intergalactic absorption models depending on vaguely known
cosmological input parameters [9]. The
extragalactic tachyon flux is not attenuated by interaction with the
background light, there is no absorption of tachyonic γ-rays.
The superluminal Proca field (1) is minimally
coupled to the electron current, and does not directly interact with
electromagnetic radiation.

The spectral curvature apparent in double-logarithmic plots of
the E2-rescaled
flux densities (16) is
intrinsic, caused by the Boltzmann factor of the electron populations
generating the tachyon flux. The curvature present in TeV γ-ray
spectra does not increase with distance, at least there is no evidence
to that effect if we compare the spectral slopes of the blazars in Fig. 1 and Fig. 2, and the
spectral maps of other flaring AGNs such as H1426+428 at z≈0.129
and 1ES 1959+650 at z≈0.047,
cf. Ref. [11]. Further
evidence for intrinsic spectral curvature is provided by Fig. 3,
depicting the spectral map of the microquasar LS 5039, a compact object
orbiting a massive O-star [32], [33] and [34]. The
spectral curvature of this Galactic binary is even more pronounced than
of the BL Lacs in Fig. 1 and Fig. 2. The same
holds true for the HESS spectral map of LS 5039 at the inferior
conjunction studied in Ref. [18].

Fig. 4 depicts
the γ-ray wideband of the TeV source HESS J1801-233
and the coincident EGRET point source 3EG J1800-2338 [35], [36] and [37]. The
extended TeV source is located on the northeastern rim of the supernova
remnant (SNR) W28, a mixed-morphology SNR interacting with molecular
clouds [38] and [39]. The
spectral map of SNR W28 in Fig. 4 is to be
compared to the unpulsed γ-ray spectrum of the Crab
Nebula, cf. Fig. 1 in Ref. [11], the
spectral map of SNR RX J1713.7-3946 in Fig. 2 of Ref. [11], and in
particular to the unidentified TeV source TeV J2032+4130 in conjunction
with the associated EGRET source 3EG J2033+4118, cf. Fig. 6 of Ref. [12]. A spectral
plateau in the MeV to GeV range occurs frequently in spectral maps of
TeV γ-ray sources, and can easily be fitted with
tachyonic cascade spectra, in contrast to electromagnetic
inverse-Compton fits. SNR W28 is located at a distance of 1.9 kpc [38]. The
thermal cascade in the MeV range (ρ2
in Fig. 4)
preceding the spectral plateau is strongly curved; the power-law slope
of the ρ1
cascade also terminates in exponential decay, but outside the presently
accessible TeV range shown in the figure. The cutoff temperature of the
nonthermal electron density generating cascade ρ1
is too high to bend the power-law slope in the TeV range covered in Fig. 4, so that
the internal energy of the shock-heated plasma could not be determined
from the χ2-fit,
in contrast with the power-law index α and the
threshold Lorentz factor γ1,
cf. caption to Table 1.

4. Conclusion

The spectral maps of two TeV γ-ray blazars
have been fitted with tachyonic cascade spectra and compared to a
Galactic γ-ray binary and supernova remnant. Table 1 contains
estimates of the thermodynamic parameters of the electron populations
generating the superluminal cascades. The spectral curvature is
intrinsic and reproduced by the tachyonic spectral densities (3) averaged
with ultra-relativistic thermal electron distributions, cf. Fig. 1, Fig. 2 and Fig. 3. The
shocked electron plasma in SNR W28 requires a nonthermal power-law
distribution to adequately reproduce the TeV cascade in Fig. 4. The
curvature in the γ-ray spectra of BL Lacs is
uncorrelated with distance, so that absorption of electromagnetic
radiation due to interaction with infrared photons is not an attractive
explanation of spectral curvature. By contrast, there is no attenuation
of the extragalactic tachyon flux, as tachyons cannot interact with
cosmic background photons, so that the observed cascades are the
intrinsic spectrum. The distance of the AGNs does not show in the
spectral curvature, as tachyonic cascades are unaffected by background
photons.

Acknowledgements

The author acknowledges the support of the Japan Society for
the Promotion of Science. The hospitality and stimulating atmosphere of
the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy,
and the Institute of Mathematical Sciences, Chennai, are likewise
gratefully acknowledged.